What is tangent law? Discuss in detail.

#### Solution

**Tangent law:**

When a magnetic needle or magnet is freely suspended in two mutually perpendicular uniform magnetic fields, it will come to rest in the direction of the resultant of the two fields.

**Explanation:**

Let B be the magnetic field produced by passing a current through the coil of the tangent Galvanometer and B_{H} be the horizontal component of the earth’s magnetic field. Under the action of two magnetic fields, the needle comes to rest making angle θ with B_{H}, such that

**Resultant position of pivoted needle**

tan θ = `"B"/"B"_"H"`

B = B_{H} tan θ ……. (1)

When no current is passed through the coil, the small magnetic needle lies along the horizontal component of Earth’s magnetic field. When the circuit is switched ON, the electric current will pass through the circular coil and produce a magnetic field. Now there are two fields that are acting mutually perpendicular to each other. They are:

- the magnetic field (B) due to the electric current in the coil acting normal to the plane of the coil.
- the horizontal component of Earth’s magnetic field (B
_{H})

Because of these crossed fields, the pivoted magnetic needle deflects through an angle θ. From tangent law, B = B_{H}tan θ When an electric current is passed through a circular coil of radius R having N turns, the magnitude of the magnetic field at the center is

B = `mu_0 "NI"/"2R"` ....(2)

From equation (1) and equation (2), we get

`mu_0 "NI"/"2R"`= B_{H}tan θ

The horizontal component of Earth’s magnetic field can be determined as

B = `mu_0 "NI"/(2 "R" tan theta)` in tesla ……. (3)